Polynomial features improve low dimensional data models with stability control
Greedy Orthogonal Polynomial Manifolds with Lipschitz Control for Dimensionality Reduction and Reduced Modeling
Machine Learning
Summary
When creating simpler versions of complex data, it helps to ensure the simplified model remains accurate even when looking at new data. The authors suggest a new way to pick parts of a polynomial-based model that are both easy to work with and keep the model from changing too wildly. They show their method can build models that are both precise and reliable beyond just the initial training data. This approach helps make better predictions and understandings outside the examples the model learned from.
What this means in practice
- •For data engineers: Build more reliable data compression tools that maintain accuracy when processing unseen data by integrating orthogonal polynomial features with controlled smoothness.
- •For simulation software developers: Develop faster reduced models for physical simulations that generalize better by limiting output sensitivity using polynomial manifold features with Lipschitz constraints.
Authors
Tomoki Koike, Benjamin Peherstorfer, Elizabeth Qian
Abstract
Polynomial manifolds provide nonlinear low-dimensional representations through polynomial decoders. We propose to represent the polynomial terms of the decoder using features that are orthonormal on the training data. This empirical orthogonality enables a greedy algorithm, Greedy-OPM, which jointly selects encoder modes and polynomial features while minimizing the reconstruction error under a prescribed bound on the decoder's Lipschitz constant. Numerical experiments show that this Lipschitz constraint improves generalization away from the training data while retaining high reconstruction accuracy.